A research blog containing 0th drafts and "open questions" - with a focus on philosophy of math.

Thursday, April 21, 2016

Three Projects Involving Dispensing With Mathematical Objects

One of the many unfortunate things about academic fashions is that when a popular project goes out of fashion, superficially similar-looking projects which don't face the same difficulties can be tarred with the same brush.Many people (myself included) feel cautious pessimism about formulating a satisfying nominalist paraphrase of contemporary scientific theories [one major issue is how to make formulate something like probability claims without invoking abstract-seeming events or propositions]. But we wouldn't want to over generalize. In this post I'm going to suggest three different motives for seeking to systematically paraphrase our best scientific theories (and many other true and false ordinary claims) in a way that dispenses with quantification over mathematical objects, and note that the requirements for success in the first (ex-fashionable) project are notably laxer in some ways than the requirements for the others.[Note: I don't mean to endorse any of the projects below, but I think that 3 and Burali-Forti based versions of 2 are at least interesting.]

Three Motivations for Paraphrasing Mathematical Objects Out of Physics

General rejection of abstracta: You deny the existence of mathematical objects because you think allowing any abstract objects are bad. (this is the classic motive)

Explaining special features of mathematical practice by rejection of mathematical objects: You deny the existence of mathematical objects because you think that not taking mathematical existence claims at face value is allows for the best account of certain special features of pure mathematical practice, (e.g., by mathematicians’ apparent freedom to choose what objects to talk in terms of/disinterest in mathematical questions that don’t effect interpretability strength, or by the Burali-Forti paradoxes in higher set theory) not to take apparent quantification over mathematical objects at face value.

Grounding math in logic/bringing out a claimed special relationship between math and logic: You may allow the existence of mathematical objects, but you’re moved by the close relationship between an intuitive modal notion of coherence/semantic consistency/logical possibility and pure mathematics to seek some kind of shared grounding and think that the coherence/logical possibility notion looks to be the more fundamental. As a result, it seems promising to seek a kind of "factoring" story, which systematically grounds all pure mathematics in facts about logical possibility, and all applied mathematics in some combo of logical possibility and intuitively non-mathematical facts.

Distinguishing these motivations/projects matters, because what you are trying to do influences what ingredients are kosher for use in your paraphrases: If you have the first motivation (defending general nominalism), you need to avoid quantifying over any other abstracta, including: platonic objects called masses or lengths which don’t come with any automatic relationship to numbers, corporations, marriages, sentences in natural languages, metaphysically possible worlds, and (perhaps) propensities. But if you have the second motivation (defending mathematical-nominalism-adopted-to-explain-special-features-of-mathematical-practice), then quantification over abstracta which don’t have the relevant special feature, (e.g., objects in domains where we don't think appear to have massive freedom to choose what objects to talk in terms of or objects which don't give rise to a version of the burali-forti paradox), is fine. And if you have the third motivation, then quantification over any objects which aren’t intuitively mathematical-- or aren't mathematical in whatever way you are claiming requires a special relationship to coherence/semantic consistency/logical possibility -- is OK.

Here's an attempted explanation for why we might care about the first project (not my favorite) for a 5 year old. I'll try to remember to come back and explain the second and third similarly.

Are there really mathematical objects like the number 3? Or are they just a figure of speech like "chips on peoples shoulders" or "the average policeman".

It might be weird if there are mathematical objects, because they would be invisible and not located anywhere in the world. Many people find the idea of such objects strange and implausible, and believing in them can seem unscientific, like believing in gods and ghosts. [this is my attempt to charitably state motivation 1]

So maybe mathematical objects are just a figure of speech.

But in this case, it should be possible to say what what we believe without using that figure of speech, without mentioning [quantifying over] mathematical objects.

However it looks very hard to do this. Specifically, it looks hard to say what we literally believe about physics without mentioning mathematical objects.

This is Quine's Indispensibility challenge to nominalists about mathematics.

Okay so my current understanding is that there is a single goal (to remove the use of mathematical objects from scientific theories) with at least three motivations. These motivations are each to some degree associated with a method for reaching the goal. One of the motivations is that mathematical objects don't exist, so shouldn't be invoked.

I suspect the unspoken reasoning there is that if entities that do not exist are invoked, a faulty pattern might be found, which becomes accepted because it explains some truths, but then is gradually munged in order to handle exceptions. In a phrase, flawed theories are based on flawed ontologies, such as the acceptance of the existence of aether, levity, or phlogiston.

I'll try to tackle this more once I get corrections to my current map ;)

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Sometimes the fastest route to getting the right answer is to say something wrong clearly enough. I'm going to try to make a lot of proposals that are initially appealing (but probably ultimately wrong) clearly.

Then (in the comments and elsewhere), I'll try to see exactly where these proposals fail. I hope you will help with the debugging!